Question

Use a normal approximation to find the probability of the indicated number of voters. In this​ case, assume that 120 eligible voters aged​ 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged​ 18-24, 22% of them voted.

Probability that fewer than 30 voted?

Answer 1

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and

p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ………….........................................................................………..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST...............................………….(1a)

Mean (average) of X = E(X) = µ = np….....................................................................……………………………………………..(2)

Variance of X = V(X) = σ² = np(1 – p)………….................................................................………………………………………..(3)

Standard Deviation of X = SD(X) = σ = √{np(1 – p)} ……......................................................…………………………………...(4)

If X ~ B(n, p), np ≥ 10 and np(1 - p) ≥ 10, then Binomial probability can be approximated by

Standard Normal probabilities by Z = (X – np)/√{np(1 - p)} ~ N(0, 1) …………………,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....,,,,,,,,,,,,,,…..(5)

Probability values for the Standard Normal Variable, Z, can be directly read off from Standard Normal Tables.................. (6a)

or can be found using Excel Function: Statistical, NORMSDIST which gives P(Z ≤ z) ….............................................……(6b)

Now, to work out the solution,

Let X = number of voters who voted in a sample of 120 eligible voters. Then, X ~ B(120, 0.22) ......................................... (7)

where 0.22 = proportion of eligible voters who vote.

By (2) and (4), mean = 26.4 and standard deviation = 4.567 ................................................................................................(8)

Probability that fewer than 30 voted

= P(X < 30)

= P[Z < {(30 – 26.4)/4.567}}   [vide Normal approximation (5), (7) and (8)]

= P(z < 0.7883)

= 0.7847 [vide (6b)] Answer

DONE